Multi-Team Bertrand Game with Heterogeneous Players

Mohammed Fathy Elettreby, Daoud Suleiman Mashat, Ashraf Mobarez Zenkour

.

DOI: 10.4236/am.2011.29164                

Abstract

In this paper, we proposed a general form of a multi-team Bertrand game. Then, we studied a two-team Bertrand game, each team consists of two firms, with heterogeneous strategies among teams and homogeneous strategies among players. We find the equilibrium solutions and the conditions of their local stability. Numerical simulations were used to illustrate the complex behaviour of the proposed model, such as period doubling bifurcation and chaos. Finally, we used the feedback control method to control the model.

Keywords

Bertrand Game, Non-convex Dynamical Multi-team Game, Incomplete Information Dynamical System, Marginal Profit Method, Nash Equilibrium

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	R. Gibbons, “A Primer in Game Theory,” Simon and Schuster, New York, 1992.
[2]	B. R. Myerson, “Game Theory: Analysis of Conflict,” Harvard University Press, Cambridge, 1997.
[3]	A. Cournot, “Researches into the Mathematical Principles of the Theory of Wealth,” Macmillan, New York, 1897.
[4]	J. Bertrand, “Theorie Mathematique de la Richesse Soaiale,” Journal des Savants, Vol. 67, 1883, pp. 499-508.
[5]	J. Nash, “Equilibrium Points in an n-Person Games,” Proceedings of the National Academy of Sciences, Vol. 36, No. 1, 1950, pp. 48-49. doi:10.1073/pnas.36.1.48
[6]	E. Ahmed, A. S. Hegazi, M. F. Elettreby and S. S. Askar, “On Multi-Team Games,” Physica A: Statistical Mechanics and Its Applications, Vol. 369, No. 2, 2006, pp. 809-816. doi:10.1016/j.physa.2006.02.011
[7]	T. Puu, “Chaos in Duopoly Pricing,” Chaos, Solitons & Fractals, Vol. 1, No. 6, 1991, pp 573-581.
[8]	T. Puu, “The Chaotic Monopolist,” Chaos, Solitons & Fractals, Vol. 5, No. 1, 1995, pp. 35-44.
[9]	M. F. Elettreby and S. Z. Hassan, “Dynamical Multi-Team Cournot Game,” Chaos, Solitons & Fractals, Vol. 27, No. 3, 2006, pp. 666-672.
[10]	E. Ahmed and A. S. Hegazi, On Dynamical Multi-Team and Signaling Games,” Applied Mathematics and Computation, Vol. 172, No. 1, 2006, pp. 524-530. doi:10.1016/j.amc.2005.02.030
[11]	S. S. Asker, “On Dynamical Multi-Team Cournot Game in Exploitation of a Renewable Resource,” Chaos, Solitons & Fractals, Vol. 32, No. 1, 2007, pp. 264-268.
[12]	E. Ahmed, M. F. Elettreby and A. S. Hegazi, “On Puu’s Incomplete Information Formulation for the Standard and Multi-Team Bertrand Game,” Chaos, Solitons & Fractals, Vol. 30, No. 5, December 2006, pp. 1180-1184.
[13]	J. X. Zhang, Q. L. Da and Y. H. Wang, “The Dynamics of Bertrand Model with Bounded Rationality,” Chaos, Solitons & Fractals, Vol. 39, No. 5, 2009, pp. 2048-2055.
[14]	D. Zhanwen, H. Qinglan and Y. Honglin, “Analysis of the Dynamics of Multi-Team Bertrand Game with Heterogeneous Players,” International Journal of Systems Science, Vol. 42, No. 6, 2010, pp. 1047-1056.
[15]	G. Gigerenzer and R. Selten, “Bounded Rationality,” MIT Press, Cambridge, 2002.
[16]	L. Edelstein-Keshet, “Mathematical Models in Biology,” Random House, New York, 1988.
[17]	E. I. Jury, “The Inners Approch to Some Problems of System Theory,” IEEE Transactions on Automatic Control, Vol. 16, No. 3, 1971, pp. 233-241. doi:10.1109/TAC.1971.1099725
[18]	E. M. Elabbasy, H. N. Agiza and A. A. Elsadany, “Analysis of Nonlinear Triopoly Game with Heterogeneous Players,” Computers & Mathematics with Applications, Vol. 57, No. 3, 2009, pp. 488-499.